A tennis championship is played on a knock-out basis (i.e. a player is out of the tournament when he loses a match).
(a) How many players participate in the tournament if 63 matches in total are played?
(b) How many matches are played in the tournament if 28 players in total participate?
Answer
The final: total matches = 1 (2 players)
Semi-finals: 2 matches (4 players) and total matches = 1 + 2 = 3 matches
Quarter finals: 4 matches (8 players) and total matches = 1 + 2 + 4 = 7 matches
Pre-quarter finals: 8 matches (16 players) and total matches = 1 + 2 + 4 + 8 = 15
Note that half the players lose their matches at each stage and are out of the tournament.
More importantly, the above explanation shows that the total number of matches played is always one less than the number of players participating in the tournament. This is basically because all players except the champion have to lose a match. One can now easily answer the two questions posed.
(a) If 63 matches are totally played, then 64 players participate in the tournament.
(b) If 28 players participate in the tournament, then 27 matches are played in total.
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